Calculating the prediction filter, viz., digital Weiner filter's autoregressive operator for accurately describing a time-invariant geophysical system is of considerable importance. Obtaining a stable and accurate solution for an ill-conditioned system of normal equations assocaited with prediction filter viz. digital Weiner filter or autoregressive operator for a time-invariant digitized data made available at smaller intervals has not been attempted. This problem is central to most measured data and is especially important in applications relating to the data available from reflection seismic, gravity, magnetic, resistivity, magnetotelluric used for prospecting for oil, mineral and water. The central problem is, therefore, to obtain a stable and accurate solution for an ill-conditioned system of normal equations assocaited with digital Weiner filter/or autoregressive operator or in general a prediction filter.
Treitel and Wang (1976) suggested a method of finding solution of an ill-conditioned system of normal equations by means of “pre-whitening”. They advocate the replacement of zero-lag term r0 of the auto-correlation coefficient sequence by a slightly larger number r′0 such that:r′0=r0+Δr0where Δr0 is small compared to r0. The replacement of r0 by r′0 results in a different system of normal equations. Treitel and Wang (1976) illustrated with examples that as the prewhitening levels used on the original ill-conditioned matrix increases, the condition number, which is a measure of ill conditioning, decreases. They observe that the percentage levels of prewhitening to be used depend, in part, on the numerical accuracy of the available computer. They also observe that more numerical computations on a given computer would be necessary if a closer estimate of such an “optimum” prewhitening level is desired. They also note, without proof that the addition of a small amount of white noise improves the conditioning of the linear system (O'Dowd, 1990).
Treitel and Wang (1976) advocate that the application of pre-whitening level to the coefficient matrix associated with an ill-conditioned system of normal equations to condition the coefficient matrix. A trade-off between the prewhitening levels and the condition numbers must be considered. Apparently, the compromise made between prewhitening level and condition number result in a system of normal equations, which may not be totally well conditioned, as some measure of ill conditioning would still persist in the system under treatment. O'Dowd (1990) observes that the application of prewhitening level is not an acceptable reason to make the change in the diagonal arbitrarily large.
Gram_Schmidt process of orthonormalisation has been used in the past to condition an ill-conditioned system of normal equations assocaited with the trend surface and to determine the trend-coefficients. Sarma and Selvaraj (1990) demonstrated the use of Gram_Schmidt process of orthonormalisation in the determination of trend-coefficients from an ill-conditioned system of normal equations assocaited with trend-surface analysis.